Geometric, electronic and magnetic structure of Fe x O + y clusters
aa r X i v : . [ phy s i c s . a t m - c l u s ] N ov Geometric, electronic, and magnetic structure of Fe x O + y clusters R. Logemann, G.A. de Wijs, M.I. Katsnelson, and A. Kirilyuk
Radboud University, Institute for Molecules and Materials, NL-6525 AJ Nijmegen, The Netherlands (Dated: August 20, 2018)Correlation between geometry, electronic structure and magnetism of solids is both intriguing andelusive. This is particularly strongly manifested in small clusters, where a vast number of unusualstructures appear. Here, we employ density functional theory in combination with a genetic searchalgorithm, GGA+ U and a hybrid functional to determine the structure of gas phase Fe x O + / y clus-ters. For Fe x O + y cation clusters we also calculate the corresponding vibration spectra and comparethem with experiments. We successfully identify Fe O +4 , Fe O +5 , Fe O +6 , Fe O +7 and propose struc-tures for Fe O +8 . Within the triangular geometric structure of Fe O +4 a non-collinear, ferrimagneticand ferromagnetic state are comparable in energy. Fe O +5 and Fe O +6 are ferrimagnetic with a resid-ual magnetic moment of 1 µ B due to ionization. Fe O +7 is ferrimagnetic due to the odd number ofFe atoms. We compare the electronic structure with bulk magnetite and find Fe O +5 , Fe O +6 , Fe O +8 to be mixed valence clusters. In contrast, in Fe O +4 and Fe O +7 , all Fe are found to be trivalent. PACS numbers: 36.40.Cg, 36.40.Mr, 61.46.Bc, 73.22.-f
In nano technology there is an ever increasing demandfor increasing the density of electronic and magnetic de-vices. This continuous downscaling trend drives the in-terest to electronic and magnetic structures at the atomicscale. In essence, two things are required: first, novelmaterials and building blocks with exotic physical prop-erties. Second, a fundamental knowledge of the physicalmechanism of magnetism at the sub-nanometer scale.Atomic clusters, having highly non-monotonous behav-ior as a function of size, are a promising model system tostudy the fundamentals of magnetism at the nanoscaleand below. Such clusters consist of only tens of atoms.Quantum mechanics starts to play an essential role atthis small scale, adding extra degrees of freedom. Sincethese clusters are studied in high vacuum, they are com-pletely isolated from their environment.To use these clusters as a model system, as a startingpoint, a detailed understanding of the relation betweentheir geometry and electronic structure is required.Even in the bulk, iron oxide has a wide variety of chem-ical compositions and phases with many interesting phe-nomena, such as the Verwey transition in magnetite.
Experiments performed on small gas phase Fe x O y clus-ters beyond the two-atom case are scarce. The structureof one and two Fe atoms with oxygen has been studiedin an argon matrix using infrared spectra. The corre-sponding vibration frequencies have been identified usingdensity functional theory (DFT).Iron-oxide nanoparticles have been investigated fortheir potential use as catalyst in chemical reactions. Fur-thermore, since the iron-oxygen interaction has a funda-mental role in many chemical and biological processes,there have been quite some studies, both experimen-tal and theoretical, of the chemical properties of Fe x O y clusters. The possible coexistence of two structural isomers forstoichiometric iron-oxide clusters in the size range n ≥ n O n and Fe n O n +1 ( n = 2-9). Furthermore, the formation of Fe x O y clus-ters has been studied in the size range ( x = 1-52). The number of theoretical studies is, however, man-ifold. The magic cluster Fe O was extensively stud-ied and identified as a cluster with C but close to D h point group symmetry. However, also the geometryand electronic structure of other cluster sizes have beenstudied theoretically.
The prediction of geometricstructures requires a systematic search of the potentialenergy surface to find the global minimum.The majority of theoretical studies were performedusing DFT.
The number of works inwhich Fe m O n clusters were studied with methods beyondDFT is very limited and restricted to very small clustersizes. For FeO + its reactivity towards H was studiedon a wave-function-based CASPT2D level. For Fe O the molecular and electronic structure were calculated us-ing both DFT and wave-function-based CCSD(T) meth-ods and a B u ground state was found. Furthermore,Ref. 25 reports that B3LYP functional and CCSD(T)calculations give the same energy ordering of differentstates, although the energy differences are overestimatedby the B3LYP approach.Recently, the structural evolution of (Fe O ) n nanoparticles was systematically investigated from theFe O cluster towards nano particles with n = 1328. In the size range of n = 1-10, an interatomic potentialwas developed and combined with a genetic algorithm insearch of the lowest-energy isomer. The isomers lowest inenergy were further optimized using DFT and the hybridfunctional B3LYP. This way, a systematic prediction ofthe cluster structure was done for neutral (Fe O ) n clus-ters.Because of its high computational burden, in DFT thegeometric structure is often only relaxed into its nearestlocal minimum on the potential energy surface (PES).There is no guarantee that this local minimum corre-sponds to the global minimum. Almost all previousworks only consider either random structures or manuallyconstructed geometries. However, for increasing clustersize these methods become less successful in finding thelowest-energy isomer. Genetic algorithms, in which sta-ble geometries are used to create new structures, provedto be efficient in finding the global energy minimum. This method has been successfully used for transition-metal oxide clusters.
Identification of the geometric cluster structure is adelicate and computationally demanding task. There-fore, comparison with an experimental method to confirmthe theoretical findings is essential. In this work, we com-bine previously reported experimental vibration spec-tra with first-principles calculations and a genetic al-gorithm to determine the geometric structure of cationicFe x O + y clusters. Of the nine cluster sizes reported inRef. 30, only the geometric structure of Fe O +6 was iden-tified. In this work, we will also identify the geomet-ric, electronic, and magnetic structure of Fe O +4 , Fe O +5 ,Fe O +7 and propose structures for Fe O +8 . I. COMPUTATIONAL DETAILS
We employ a genetic algorithm (GA) as is described inRef. 27 in combination with DFT to optimize the clusterstructures. For this we use the Vienna ab-initio simu-lation package ( vasp ) using the projector augmentedwave (PAW) method. Since the geometry optimiza-tion is the most computationally expensive part of thegenetic algorithm, we use the PBE+ U method withlimited accuracy for the genetic algorithm. For all ob-tained isomers low in energy, we reoptimized the geo-metric structure using the hybrid B3LYP functional withhigher accuracy and consider different magnetic config-urations. We then calculate the vibration spectra andcompare them with experimental results.Within the DFT framework, functionals based on thelocal density approximation (LDA) or general gradientapproximation (GGA) fail to describe strongly interact-ing systems such as transition-metal oxides. Due tothe overestimation of the electron self-interaction, theypredict metallic behavior instead of the (correct) wide-band-gap insulator. In an attempt to correct for thisself-interaction, one can, for example, employ a hybridfunctional, where a typical amount of 20% of Hartree-Fock energy is incorporated into the exchange-correlationfunctional. Especially for the B3LYP functional it hasbeen shown that this results in good agreement be-tween the geometric structure and vibrational spectrafor clusters.
However, hybrid functionals are quitecomputationally expensive compared to LDA and GGAfunctionals. Therefore, in the genetic algorithm we em-ploy the GGA+ U method to take into account that FeOclusters are strongly interacting systems. We use therotational invariant implementation introduced by Du-darev and a plane wave cutoff energy of 300 eV for thesecalculations. The differences between GGA and GGA+ U for iron-oxide cluster calculations have been analyzed in Ref. 15.This study stresses the importance to go beyond GGA fortransition-metal oxide clusters calculations. Aside fromthe well-known difference for the electronic and magneticstructure, it even finds a different lowest energy isomerthan GGA for Fe O . In our genetic algorithm cal-culations we use an U eff = U − J of 3 eV for the Featoms, based on a comparison between B3LYP calcula-tions and PBE+ U calculations for the smallest cluster,Fe O (see Sec. II B). For this comparison we also cal-culated the mean absolute difference (∆) between theoccupied Kohn-Sham energies ( E i ) using B3LYP andPBE+ U : ∆ = n X i =1 | E PBE+ Ui − E B3LYP i | n , (1)where n is the number of occupied Kohn-Sham levels.Note that, the binding distances are only weakly depen-dent on the used U eff and our value of 3 eV is close to val-ues used in other works (e.g., 5 eV , 3.6 eV , 3.6 eV ).We used the genetic algorithm as described in detailin Ref. 27. New geometries are formed by the Deaven-Ho cut and splice crossover operation. To determine thefitness we used an exponential function. A generationtypically consists of 20 clusters. It has been shown thatthe geometry of Fe x O y clusters only weakly depends onthe magnetic degree of freedom. Therefore, we restrictourselves to the ferromagnetic case in our genetic algo-rithm.For all obtained isomers low in energy, we reopti-mized the geometric structure using the hybrid B3LYPfunctional and consider all possible collinear ori-entations of the Fe magnetic moments by constrainingthe difference in majority and minority electrons. Allforces were minimized below 10 − eV/˚A. Standard rec-ommended PAWs with an energy cutoff of 400.0 eV areused. The clusters are placed in a periodic box of asize between 11 and 17 ˚A, which we checked to be suf-ficiently large to eliminate inter cluster interactions foreach cluster size. For the cluster calculations, a single k -point (Γ) is used. Since we also consider cationic clus-ters, a positive uniform background charge is added andwe correct the leading errors in the potential. Allsimulations were performed without any symmetry con-straints. The reported symmetry groups are determinedafterwards within 0.03 ˚A. For the density of states (DOS)calculations we used a Gaussian smearing of 0.1 eV forvisual clarity.To obtain the vibration spectra, the Hessian matrixof an optimized geometry is calculated by consideringfinite ionic displacements of 0.015 ˚A for all Cartesian co-ordinates of each atom. The vibration frequencies areobtained by diagonalization of the Hessian matrix. Theabsorption intensity A i is calculated using A i = 974 . g i (cid:18) ∂µ∂Q i (cid:19) , (2)where g i is the degeneracy of the vibration mode, Q i the mass weighted vibrational mode, µ the electric dipolemoment, and 974.86 an empirical factor. A method basedon four displacements for each ion was also tested butyielded the same frequencies and absorption intensities.Zero-point vibrational energies (ZPVE) were calculatedfor the isomers lowest in energy of which the vibrationspectra are also shown.For a quantitative comparison between experimen-tal and calculated vibrational spectra, we calculate thePendry’s reliability factor. The Pendry’s reliability fac-tor is a well-established method in low-energy electrondiffraction (LEED) to quantify the agreement in contin-uous spectra and has also been applied to vibrationalspectroscopy. The experimental used infrared multiphoton dissoci-ation method (IR-MPD) does not only depend on theabsorption cross section of a vibrational mode, but alsoon the dissociation cross section. Therefore, we use thePendry’s reliability factor to quantify the comparison ofvibration spectra since it is mainly sensitive to peak posi-tions opposed to a comparison of squared intensity. Thispeak sensitivity is achieved by comparing the renormal-ized logarithmic derivative of the intensity I ( ω ): Y ( ω ) = L − ( ω ) L − ( ω ) + W , (3)where L ( ω ) = I ′ ( ω ) /I ( ω ) and W is the typical FWHM ofthe peaks in the spectra. The Pendry’s reliability factoris defined as: R P = Z (cid:2) Y th ( ω ) − Y expt ( ω ) (cid:3) Y ( ω ) + Y ( ω ) dω, (4)where we integrate over the experimental range of fre-quencies. R P values range from 0 to 2, where 0 meansperfect agreement, 1 uncorrelated spectra, and 2 per-fect anticorrelation. In practice, R P values of 0.3 areconsidered acceptable agreement within LEED. Y ( ω ) isstrongly dependent on experimental noise and valuesclose to zero, hence, we calculate Y expt ( ω ) by fitting theexperimental spectrum with multiple Lorentzian peaksand extract the corresponding W . The theoretical fre-quencies are also convoluted with Lorentzian peaks withthe same W . R P is always minimized as function of arigid shift of all theoretical frequencies.For the calculations on magnetite we used the vasp code. We used a Monkhorst grid of 6 × × U implementation by Lichtenstein et al. with effective on-site Coulomb and exchange parameters: U = 4 . and J = 0 .
89 eV for the Fe ions.We used the monoclinic structure as described inRefs. 39,49, and calculated the electron density with 56atoms in the unit cell. In Ref. 39, the charge and mag-netic moment were calculated by integrating the densityand spin density in a sphere with a radius of 1 ˚A for Fe. This radius appears to be chosen such that compara-ble values with neutron and x-ray diffraction experimentswere obtained.Note, there is no unambiguous way to define theseradii in systems consisting of two or more atom types.Therefore, we checked the correspondence of our resultsto the earlier reported ones and also performed calcula-tions with a larger radius of 1.3 ˚A for Fe and 0.82 ˚A for O.This is a reasonable choice for Fe m O + n clusters since theoverlap between different spheres is minimal, but most ofthe intra cluster space is covered. II. RESULTS AND DISCUSSIONA. Magnetite
Even in the bulk, iron oxide is well known for its widevariety of phases and transitions. Magnetite (Fe O ), themost stable phase of Fe m O n , is for example well knownfor its Verwey transition. Above the transition temper-ature T V , the structure is a cubic inverse spinel. Uponcooling below T V , the conductivity decreases by two or-ders of magnitude due to charge ordering. Furthermore,the structure changes to monoclinic.Magnetite has the formal chemical formula(Fe A [Fe , Fe ] B O ) where tetrahedral A sitesare occupied by Fe and B sites contain both divalent(Fe ) and trivalent (Fe ) iron atoms. Since magnetiteis a mixed valence system, it is an excellent referencesystem for our cluster calculations to determine theirvalence state and corresponding magnetic moment. TABLE I: Spin moments within atomic spheres of 1.3 ˚A forthe Fe ions in monoclinic Fe O . For reference the valueswithin a sphere of 1.0 ˚A are also shown. A and B labels areconsistent with Ref. 39.Site Spin moment ( µ B ) Spin moment ( µ B )Radius sphere 1 . . (A) − . − . (B1) 3 .
69 3 . (B2) 4 .
15 3 . (B3) 4 .
06 3 . (B4) 3 .
64 3 . In Table I, the spin moments are shown for the dif-ferent iron ions. The magnetic moments on the A and B sites are antiparallel creating a ferrimagnetic struc-ture. Within the atomic spheres of 1.3 ˚A the Fe andFe ions have a distinct magnetic moment of 4.0 µ B and 3.7 µ B respectively. Note the difference of 0.3 µ B ismuch smaller than the 1 µ B atomic value and does notdepend on the size of the atomic sphere used in the rangebetween 1.0 and 1.3 ˚A. B. GGA+U
To determine the optimal U eff in comparison to theB3LYP functional for the genetic algorithm, we per-formed PBE+ U calculations on the neutral Fe O clus-ter. The results for the electronic DOS are shown inFig. 1 and compared with the hybrid B3LYP functional. (cid:1)=B==U8nt=(cid:8)(cid:9) [(cid:8)1[(cid:8)S=38st (cid:1) =[(cid:8)1(cid:15) f S=f8yi= (cid:1) =[(cid:8)1(cid:15) S=38UU= (cid:1) = (cid:19) (cid:8)(cid:20)(cid:20) =B=n(cid:8)(cid:9) (cid:1)=B==U8nU=(cid:8)(cid:9) [(cid:8)1[(cid:8)S=38sf (cid:1) =[(cid:8)1(cid:15) f S=f8ys= (cid:1) =[(cid:8)1(cid:15) S=38UU= (cid:1) = (cid:19) (cid:8)(cid:20)(cid:20) =B=D(cid:8)(cid:9) (cid:1)=B==U8ni=(cid:8)(cid:9) [(cid:8)1[(cid:8)S=38sU (cid:1) =[(cid:8)1(cid:15) f S=f8ys= (cid:1) =[(cid:8)1(cid:15) S=38UU= (cid:1) = (cid:19) (cid:8)(cid:20)(cid:20) =B=3(cid:8)(cid:9) H (cid:8)(cid:23) (cid:24) (cid:25) (cid:26) (cid:27) = (cid:28) (cid:20)= (cid:29) (cid:26) (cid:30) (cid:26) (cid:8) (cid:24) (cid:1)=B==U8sn=(cid:8)(cid:9) (cid:19) (cid:8)(cid:20)(cid:20) =B=f(cid:8)(cid:9) [(cid:8)1[(cid:8)S=38sU (cid:1) =[(cid:8)1(cid:15) f S=f8yn= (cid:1) =[(cid:8)1(cid:15) S=38UU= (cid:1) =(cid:1)=B==U8it=(cid:8)(cid:9) (cid:19) (cid:8)(cid:20)(cid:20) =B=U(cid:8)(cid:9) [(cid:8)1[(cid:8)S=38sU (cid:1) =[(cid:8)1(cid:15) f S=f8yn= (cid:1) =[(cid:8)1(cid:15) S=38UU= (cid:1) =[(cid:8)1[(cid:8)S=38sf (cid:1) =[(cid:8)1(cid:15) f S=f8yn= (cid:1) =[(cid:8)1(cid:15) S=f8oo= (cid:1) = dD!" [(cid:8)=(cid:24)[(cid:8)=$[(cid:8)=%(cid:15)=(cid:24)(cid:15)=$ M1M −(cid:15)((cid:15) =)(cid:8)(cid:9)*= +fU +t8s +s +38s U 38s s
FIG. 1: (Color online) The density of states for the hybridB3LYP functional and PBE+ U for different values of U eff .The average inter atomic distances are shown on the right,where Fe-O and Fe-O refer to the Fe-O distances betweenbridging O atoms (side) and the capping O atom (center),respectively. The mean absolute difference ∆ [Eq. 1] betweenthe PBE+ U and B3LYP energy levels is also shown and isminimal for U eff = 3 eV, indicating the best match in DOS. The valence states within -4 and 0 eV are formed byhybridized orbitals between the d orbitals of iron andthe p orbitals of oxygen. For increasing U , the majorityspin d orbitals of Fe decrease in energy, whereas HOMO-LUMO gap increases. Note that the HOMO-LUMO gapof 1.5 eV for U eff = 4 eV still is 0.9 eV smaller than the2.4 eV gap for B3LYP. Furthermore, for U eff = 2 and3 eV the Fe d DOS features are very similar to those ofthe B3LYP result. To quantify this we also calculatedthe mean absolute difference ∆ [Eq. 1] for the occupiedlevels; the results are shown in Fig. 1. ∆ is minimalfor U eff = 3 eV, indicating the best DOS correspondenceto B3LYP. We also show the corresponding bonding dis-tances within the cluster, where Fe-O and Fe-O referto the Fe-O distances between bridging O atoms (side)and the capping O atom (center), respectively. Note theinteratomic distances only change very little with increas-ing U eff . For U eff = 3 eV, the binding distances are within 0.01 ˚A; furthermore, for U eff = 3 eV and B3LYP the oc-cupied d orbitals of Fe are at comparable energies with re-spect to the HOMO level. We therefore used U eff = 3 eVfor our genetic algorithm calculations. C. Fe O Although the possible number of isomers increasesrapidly with cluster size, for small systems such as Fe O the number of possibilities is still small. In Fe O , the Featoms can either form a triangle or a chain. For the trian-gular configuration, two isomers are low in energy. Thefirst isomer consists of a ring like structure where the Oatoms occupy bridging states and one O atom caps the Fetriangle as is shown in Fig. 2 (a) . In the second isomer,the additional O atom is not located above the centerbut forms an extra bridge between the two ferromagnetic(FM) ordered Fe atoms as is shown in Fig. 2 (b) . (cid:1)(cid:2)(cid:3) (cid:9) (cid:10)(cid:3) (cid:11) (cid:12) (cid:13) e(cid:14) (cid:3) (cid:4) (cid:15) (cid:20)(cid:21)(cid:22)(cid:10)e(cid:8)(cid:23)(cid:12)(cid:10)(cid:3)(cid:24)(cid:22)(cid:25)(cid:23)(cid:24)(cid:22)(cid:26)(cid:10)e(cid:14)(cid:27) (cid:28) (cid:15) (cid:4)(cid:5) (cid:6) (cid:7) (cid:8)(cid:9) FIG. 2: (Color online) The energy as function of spin mag-netization for different neutral Fe O isomers. The geometricfigures on the right show the corresponding geometric struc-ture. O atoms are shown in red, Fe spin up and Fe spindown are indicated with orange (red) and green (blue) colors(arrows), respectively. For the lowest magnetic states the rel-ative energy differences are also shown in black. Isomers (a) (black line) and (b) (red line) are equally low in energy witha ferrimagnetic and ferromagnetic ground state, respectively(0 eV). The M = 6 µ B state of isomer (a) is 14 meV higherin energy. Figure 2 shows the energy as a function of spin mag-netic moment for the neutral Fe O cluster with fourdifferent isomers. For all spin magnetizations, the ge-ometric structure is optimized and shown on the rightwith its magnetic structure lowest in energy. In Fig. 2and the rest of this work, Fe spin up and Fe spin down areindicated with orange (red) and green (blue) colors (ar-rows), respectively. O atoms are shown in red. For theneutral cluster, the two triangular isomers are equallylow in energy with two different magnetic configurations.The difference is smaller than 1 meV and therefore be-yond the accuracy of DFT. In isomer (a) , as indicated bythe black line in Fig. 2, the magnetic ground state corre-sponds to ferromagnetic alignment between the magneticmoments on the Fe atoms and a total magnetic momentof 14 µ B . The Fe-Fe distances are 2.51 ˚A, the Fe-O dis-tances for the bridging O atoms and capping O atom are1.84 and 1.99 ˚A, respectively. Aside from the FM groundstate, also the ferrimagnetic state with a spin magneti-zation of 4 µ B is low in energy and only 14 meV higherthan the ferromagnetic state. Note we also considereda noncollinear magnetic state with M = 0 µ B , but thismagnetic configuration did not turn out to be energeti-cally stable.Isomer (b) is equally low in energy and shown in redin Fig. 2. The magnetic ground state corresponds to aferrimagnetic alignment where the two ferromagneticallyaligned Fe atoms have Fe-O-Fe angles of approximately90 ◦ .We also considered zero point vibrational energies forthe three lowest-energy levels. When we include theseinto our consideration, the ferromagnetic state, indicatedby the black line, is lowest in energy, and the M = 4 µ B and M = 6 µ B states are 17 and 19 meV higher in energy,respectively. D. Fe O +4 For the cation Fe O +4 cluster we also considered ringand chain configurations with different oxygen locations.For all four isomers we calculated all possible differentcollinear magnetic states. Since an antiferromagnetic(AFM) triangle is the most simple example of geometri-cally frustrated magnetism, we also considered the non-collinear state with M = 0 µ B where all magnetic mo-ments have 120 ◦ angles with respect to each other. Theresults are shown in Fig. 3. For the charged Fe O +4 clus-ter, the isomer with a Fe triangle where the fourth Oatom caps the triangle is, like in the neutral cluster, low-est in energy, as is shown in Fig. 4. Three magnetic statesare low in energy: 0, 5 and 15 µ B , with the M = 5 µ B state being lowest in energy, and the non-collinear 0 µ B and ferromagnetic 15 µ B are 20 meV and 58 meV higherin energy respectively.The ferrimagnetic state which is lowest in energy, hasa reduced symmetry ( C v ) with respect to the ferromag-netic state ( C v ) and the antiferromagnetic state. Thiscould indicate a Jahn-Teller distortion, but could alsobe the result of the inability of DFT to correctly modelthe antiferromagnetic ground state. However, to dis-tinguish between these two cases, methods beyond DFTsuch as CASPT2 and CCSD(T) are required and there-fore beyond the scope of this work. Note that differentmagnetic states only lead to minor differences in the vi-brational frequencies.Interestingly, the typical classical displacement duringa zero-point vibration in these clusters is of the orderof 0.03 ˚A. This is of the same order as the typical dif- ytµ8(cid:5)(cid:6)(cid:7)yp18(cid:5)(cid:6)(cid:7) 18(cid:6)(cid:7) (cid:10) (cid:11)(cid:6) (cid:12) (cid:13) (cid:14) (cid:6) (cid:7) (cid:16) (cid:20)(cid:21)(cid:22)(cid:11)8(cid:5)(cid:23)(cid:13)(cid:11)(cid:6)(cid:24)(cid:22)(cid:25)(cid:23)(cid:24)(cid:22)(cid:26)(cid:11)8(cid:15)(cid:27) (cid:28) (cid:16) (cid:1)(cid:2)(cid:3) (cid:4)(cid:5) (cid:6) (cid:7) (cid:8) (cid:9)
FIG. 3: (Color online) Energy of the Fe O +4 isomers as func-tion of spin magnetization. Figures on the right indicate thecorresponding structure. The isomer lowest in energy (a) isa Fe triangle with three bridge O atoms and one O atom cap-ping the triangle. For this isomer, the ferrimagnetic 5 µ B stateis lowest in energy. The antiferromagnetic 0 µ B and ferromag-netic 15 µ B state are 20 and 58 meV higher in energy, respec-tively. Note the antiferromagnetic 0 µ B state corresponds to anon-collinear orientation with 120 ◦ angles between the spins.FIG. 4: (Color online) The neutral (left) and cation (right)Fe O lowest-energy isomers. Fe spin up and Fe spin down areindicated with orange (red) and green (blue) colors (arrows),respectively. O atoms are shown in red. The interatomicdistances are shown in black. The neutral and cation clusterhave C v and C v point group symmetry, respectively. ference in inter atomic distances between different mag-netic states. Therefore, this could lead to interesting phe-nomena in which, for example, there is a strong couplingthrough exchange between vibrations and magnetism.The second triangular isomer of Fe O +4 is 154 meVhigher in energy and also consists of a ring structure. Themagnetic state lowest in energy has a magnetic momentof 5 µ B . The Fe-Fe bonding distances are 2.5 and 3.0 ˚Abetween the AFM and FM bonds within the structure.The Fe-O distances vary between 1.7 and 1.9 ˚A. Theisomer has a C v point group symmetry.The third and fourth isomers consist of a linear chainof Fe atoms with two O bridging atoms between each Fepair. The two planes can be parallel or perpendicular, (cid:1)(cid:2) (cid:3) (cid:4) (cid:5)(cid:6) (cid:1) (cid:2) [ I1pbViio (cid:12)(cid:13)(cid:14) (cid:7) (cid:1) (cid:2) [ I1rnVasb (cid:12)(cid:13)(cid:14) (cid:8) (cid:1) (cid:2) [ I1sI (cid:19) (cid:1) (cid:20)(cid:21) (cid:22) (cid:23) (cid:24) (cid:25) (cid:26) (cid:27) (cid:23)(cid:28) (cid:9)(cid:10)(cid:11)(cid:12)(cid:13) (cid:29)(cid:20)(cid:30)(cid:13)(cid:28)(cid:31)(cid:12)(cid:21)(cid:13)(cid:24) !(cid:12) oII pII iII aIII abII FIG. 5: (Color online) The experimental vibration spectraof Fe O +4 and the calculated isomers lowest in energy. Thereported energy differences include ZPVE. The Pendry’s reli-ability factor [Eq. 4] is also shown for each isomer. where the latter is lower in energy. Both isomers have amagnetic moment of 5 µ B .In Fig. 5, both the experimental and calculated vibra-tion spectra for the different isomers are shown. Theexperimental spectrum consists of three peaks at 540,610 and 670 cm − . The best match is given by isomer (a) with calculated vibrations at 505, 630 and 660 cm − and a corresponding lowest- R P factor of 0 .
30, indicat-ing a reasonable match with the experimental spectrum.Since isomer (a) is also the lowest in energy, it is identi-fied as the experimentally observed structure.
E. Fe O / +5 Fe O also consists of a ring structure in which theO atoms occupy the bridging sites and one O atom islocated above the center, as is shown in Fig. 6. The clus-ter has antiferromagnetic order. However, not all Fe-Febonds are antiferromagnetic, but also two ferromagnet-ically aligned bonds are present. Therefore, the clusterhas no C v point group symmetry but C , since Fe-Feand Fe-O distances vary between 2.72-2.74 ˚A and 1.79-2.33 ˚A respectively. The magnetic state with four AFMFe-Fe bonds is 308 meV higher in energy.For Fe O +5 the isomer lowest in energy consists of thesame ring structure but is more symmetry broken, sincethe O atom above the ring is off-center as is shown inFig. 6. Therefore the two Fe-Fe distances are 2.69 and3.07 ˚A, the Fe-O distances vary between 1.76 and 2.01 ˚A.The isomer has C s point group symmetry. Two Fe O squares are present within the cluster. Isomer (a) hasa magnetic moment of 1 µ B due to ionization. Interest-ingly, the ionized cluster has a different magnetic groundstate with four AFM Fe-Fe bonds opposed to the neutralcluster. FIG. 6: (color online) The neutral (left) and cation (right)Fe O lowest energy isomers. The neutral cluster has C sym-metry, whereas the cation cluster has C s symmetry. In Fig. 7 (b) , we also show the vibration spectrum ofthe ferromagnetic state of this cluster. The Fe-Fe dis-tances are increased to 2.74 and 3.11 ˚A, respectively. Theferromagnetic structure is 514 meV higher in energy. Thevibration spectrum is similar but slightly shifted to theblue due to the increased bonding distances. (cid:1) (cid:2) - bIisV pvp (cid:12)(cid:13)(cid:14) (cid:1) (cid:1) (cid:2) - bIttV ptv (cid:12)(cid:13)(cid:14) (cid:2) (cid:1) (cid:2) - bItbV tsp (cid:12)(cid:13)(cid:14) (cid:3) (cid:1) (cid:2) - bIpo (cid:17) (cid:1) (cid:18)(cid:19) (cid:20) (cid:21) (cid:22) (cid:23) (cid:24) (cid:25) (cid:21)(cid:26) (cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:27)(cid:18)(cid:28)(cid:13)(cid:26)(cid:29)(cid:12)(cid:19)(cid:13)(cid:22) (cid:30)(cid:31)(cid:12) ! obb pbb ibb Wbb sbbb sobb (cid:9)(cid:10) (cid:11) (cid:12) (cid:13)+ FIG. 7: (Color online) The experimental and calculated vibra-tion spectra of Fe O +5 . The isomer shown in (a) is both thelowest in energy and R P [Eq. 4] and can therefore be iden-tified as the experimentally observed geometrical structure.The reported energy differences include ZPVE. The second isomer, 459 meV higher in energy, is shownin Fig. 7 (c) . This cage-like structure has C v point groupsymmetry and a magnetic moment of 9 µ B . Figure 7 (d) shows the third isomer which is 494 meV higher in en-ergy compared to Fig. 7 (a) . The isomer has almost nosymmetry ( C ), and consists of a ring where one Fe-Febond has two bridging O atoms. The Fe-Fe binding dis-tances vary between 2.62 and 3.13 ˚A. The isomer has amagnetic moment of 1 µ B .In the experimental vibration spectrum of Fe O +5 shown in Fig. 7, five vibration frequencies can be ob-served: 450, 615, 760, 810, and 1070 cm − . The vibra-tion at 1070 cm − can be identified as a shifted vibrationin the O messenger attached to the cluster-messengercomplex and is therefore omitted in the R P calculation. The best fit is given by isomer Fig. 7 (a) with R P = 0 . − match allwithin 30 cm − to the experimental spectrum. Also, therelative intensities between different vibrations are verysimilar. Although the ferromagnetic order increases thebinding distances within the cluster, the changes in thevibration spectrum of Fig. 7 (b) are small and thereforethe structure corresponding to Figs. 7 (a) and 7 (b) can beidentified as the experimentally observed structure andthe IR-MPD method is not able to resolve the magneticstate in this case. F. Fe O / +6 In Ref. , the Fe O +6 cluster was already identified asthe structure shown in Fig. 8 (b) . The reported magneticstructure was ferrimagnetic with a magnetic moment of9 µ B . (cid:1) (cid:2) - bI rvesWn (cid:13)(cid:14)(cid:15) (cid:1) (cid:1) (cid:2) - bIpW (cid:17) (cid:1) (cid:18)(cid:19) (cid:20) (cid:21) (cid:22) (cid:23) (cid:24) (cid:25) (cid:21)(cid:26) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6) (cid:27)(cid:18)(cid:28)(cid:14)(cid:26)(cid:29)(cid:13)(cid:19)(cid:14)(cid:22) (cid:30)(cid:31)(cid:13) ! obb pbb ibb Wbb sbbb sobb (cid:7)(cid:8) (cid:9) (cid:10) (cid:11)(cid:12) FIG. 8: (color online) The experimental and calculated vibra-tion spectra of Fe O +6 for both the previous and new magneticground state. The vibration frequencies are very similar butdiffer in absorption intensity. The M = 1 µ B state in (a) is187 meV lower in energy. In our calculations a magnetic state lower in energywas found for the same geometric structure for bothFe O and Fe O +6 . In this state Fe O and Fe O +6 have amagnetic moment of 0 and 1 µ B respectively as is shownin Fig. 9. These structures are 194 and 187 meV lowerin energy for Fe O and Fe O +6 in comparison to thepreviously reported state. The antiferromagnetic mag-netic ground state of Fe O was also previously reportedin Ref. 26. For Fe O we also calculated a noncollinearstate where all magnetic moments point towards the cen-ter of mass, such state with M = 0 µ B is 30 meV higherin energy compared to the collinear M = 0 µ B state.For the neutral cluster, minima in energy are obtainedfor M = 0, 10, 20 µ B corresponding to flips of atomic (cid:1)(cid:2) t (cid:4) z1 (cid:1)(cid:2) t (cid:4) z[ (cid:8) (cid:9)(cid:2) (cid:10) (cid:11) (cid:12) (cid:2) (cid:15) (cid:16) (cid:28) (cid:16)1 M t z µ 21 2M 2t 2z 2µ M1 (cid:1)(cid:2) t (cid:4) z15[ FIG. 9: (color online) Energy as function of magnetization ofthe neutral Fe O and cationic Fe O +6 clusters. The magneticground state corresponds to a total spin magnetic moment of M = 0 and M = 1 µ B for Fe O , and Fe O +6 respectively. magnetic moments of 5 µ B for each Fe atom. Note thisalso matches with an ionic picture in which the Fe atomsin Fe O have a Fe valence state resulting in an atomicmagnetic moment of 5 µ B . The corresponding structureis shown in Fig. 10. In Ref. 30 is mentioned that thesymmetry in the M = 10 µ B state is reduced from T d forthe ferromagnetic state to C v . In this antiferromagneticground state, the neutral cluster has D d symmetry. InFe O +6 the symmetry is reduced even further to C s as isshown in Fig. 10. FIG. 10: (Color online) The neutral (left) and cation (right)Fe O lowest energy isomers. The neutral cluster has D d symmetry, whereas the cation cluster has C s symmetry. Figure 8 shows both calculated and experimental spec-tra for Fe O +6 . The vibration spectra for the two calcu-lated magnetic states in Figs. (a) and 8 (b) show verysimilar behavior. The R P values of isomer Fig. 8 (a) (0.48) and Fig. 8 (b) (0.39) are both large and indicate abetter match for isomer Fig. 8 (b) . Although the spectrafor Figs. 8 (a) and 8 (b) are very similar, the ferrimag-netic structure has an extra vibration at 720 cm − withsmall IR absorption. Furthermore, around 550 cm − ,vibrations differ slightly in frequency. Since the men-tioned differences cannot be experimentally resolved, theIR-MPD method is unable to resolve between differentmagnetic states and another type of experiments suchas Stern-Gerlach deflection is required to determine themagnetic moment. G. Fe O / +7 The neutral Fe O cluster has a “basket” geometryas is shown in Fig. 11. The magnetic ground state isferrimagnetic with a total moment of 4 µ B due to theodd number of Fe atoms. The cluster has C v symmetry. FIG. 11: (Color online) The neutral (left) and cation (right)Fe O lowest-energy isomers. The neutral cluster has C v symmetry, whereas the cation cluster has no symmetry. The cationic structure of Fe O +7 is very different andshown in Fig. 11. Like Fe O +6 , it consists of a cage-likestructure. The Fe-Fe distances range from 2.7 to 3.1 ˚A.Except for the triple bound O atom, all O atoms formbridges between two Fe atoms. The ground state has amagnetic moment of 5 µ B . The second isomer is similar (cid:1) (cid:2) o vWc[ (cid:1) (cid:2) o vWrb (cid:2) (cid:2) o vWc[ (cid:3)(cid:4)(cid:5)(cid:6)(cid:7) (cid:17)(cid:18)(cid:19)(cid:11)(cid:20)(cid:21)(cid:16)(cid:22)(cid:11)(cid:23) (cid:24)(cid:25)(cid:16) mn (cid:27)uvv rvv cvv ]vv nvvv nuvv (cid:30) (cid:1) (cid:18)(cid:22) (cid:31) (cid:23) ! " (cid:20) (cid:8) (cid:9) (cid:10) (cid:11) (cid:12)(cid:13) FIG. 12: (Color online) The experimental and calculated vi-bration spectra of Fe O +7 . The reported energy differencesinclude ZPVE. to the neutral ”basket” structure and is 394 meV higherin energy as is shown in Fig. 12 (b) . The structure has C s symmetry and a magnetic moment of 5 µ B . However,the atomic spin moments have a different arrangementfor the neutral and cationic state.The third isomer is shown in Fig. 12 (c) and is 1.04 eVhigher in energy. It contains two triple bonded O atomsand is ferrimagnetic with M = 5 µ B .The experimental vibration spectrum shown in Fig. 12 has eight distinct vibrations at 375, 490, 520, 570, 615,710, 780, and 830 cm − which are best resembled by theisomer lowest in energy shown in Fig. 12 (a) , althoughthe gap between 615 and 710 cm − seems to be underesti-mated. Note that this also explains the high- R P factor of0.65 for isomer Fig. 12 (a) . Similar to Fe O +5 and Fe O +6 the absorption intensities of vibrations in the range of300-500 cm − are systematically underestimated. Theindividual vibrations of isomer Fig. 12 (a) are all in agree-ment within 35 cm − . Although isomer Fig. 12 (b) has alower R P = 0.43, the energy difference of 407 meV withisomer Fig. 12 (a) is large and isomer Fig. 12 (b) has avibration at 450 cm − which is not present in the exper-imental spectrum and lacks the experimental 375 cm − vibration. Therefore, isomer Fig. 12 (a) can be identifiedas the most probable ground state. H. Fe O +8 The isomer lowest in energy found for Fe O +8 is shownin Fig. 13 and has C s symmetry where the reflection planeis located through Fe atoms 1, 3, and 6. The magneticmoment of this isomer is 1 µ B . FIG. 13: (Color online) The cation Fe O +8 isomer lowest inenergy. The cluster has C s symmetry. The second isomer low in energy is shown in Fig. 14 (b) .In this isomer no symmetry is present. Compared to thelowest found isomer in Fig. 14 (a) it is 413 meV higher inenergy and also has a magnetic moment of 1 µ B .Figure 14 (c) shows the third isomer, which is a dis-torted octahedral of Fe atoms in which the O atoms capthe Fe triangles. The structure is slightly distorted dueto the AFM order between spins, which lead to slightlyaltered Fe-Fe distances. This isomer is 483 meV higherin energy than isomer Fig. 14 (a) .Figure 14 also shows the corresponding vibration spec-tra of the mentioned isomers and the experimental spec-trum. The experimental spectrum has vibrations at 392,420, 500, 730 and 763 cm − . Note that none of theprovided isomers match the experimental vibration spec-trum completely. This is also shown by the large- R P values of 0.56-0.61 for all calculated isomers. The isomerlowest in energy Fig. 14 (a) is the best match since it alsohas vibrations at 420 and 500 cm − , but the vibrations at (cid:1) (cid:2) o a1r] (cid:1) eb-u (cid:13)(cid:14)(cid:15)(cid:1) (cid:2) o a1[v (cid:2) ebvu (cid:13)(cid:14)(cid:15)(cid:1) (cid:2) o a1r[ (cid:3)(cid:4)(cid:5)(cid:6)(cid:7) (cid:18)(cid:19)(cid:20)(cid:14)(cid:21)(cid:22)(cid:13)(cid:23)(cid:14)(cid:24) (cid:25)(cid:26)(cid:13) (cid:28)naa baa [aa -aa vaaa vnaa (cid:30) (cid:1) (cid:19)(cid:23) (cid:31) (cid:24) ! " (cid:21) (cid:8) (cid:9) (cid:10) (cid:11) (cid:12)(cid:13) FIG. 14: (color online) The experimental and calculated vi-bration spectra of Fe O +8 . The isomer shown in (a) is the low-est in energy. The reported energy differences include ZPVE.
804 and 825 are considerably shifted with respect to 730and 763 cm − . Furthermore, the vibrations at 640, 671,and 713 cm − are not present in the experimental spec-trum. The vibration spectra shown in Figs. 14 (b) and14 (c) fit even worse. Therefore, we can not successfullyidentify the Fe O +8 structure.Note that our genetic algorithm implementation onlyuses geometry optimization at the DFT level. At clustersizes of Fe O +8 and larger, preselection using empiricalpotentials instead of immediate geometry optimizationusing DFT might be more efficient in generating possibleisomers. I. Electronic structure
In the bulk, iron-oxide materials have many differentcrystal structures such as hematite, wustite, and mag-netite with all corresponding different electronic struc-tures. While in hematite only trivalent Fe is present,the mixed valence state (Fe A [Fe , Fe ] B O ) in mag-netite leads to interesting physical phenomena such asferrimagnetic ordering between the sublattices A and B and the Verwey transition in which orbital orderingleads to a first-order phase transition in the electricalconductivity. In clusters, stoichiometries corresponding to bothhematite (Fe O ) and magnetite (Fe O , Fe O ) andother combinations (Fe O , Fe O ) occur. We thereforeexpect divalent and trivalent Fe cations to be present inthe reported clusters. There is no unique method to de-termine the valence state in materials consisting of mul-tiple types of elements. We therefore compare both thelocal magnetic moments and the local density of states(LDOS) for our cluster calculations with bulk magnetiteresults shown in Section II A. Since the Fe and Fe features in the LDOS are very similar for different cluster sizes, we show the LDOS of Fe O +5 which contains bothFe and Fe in Fig. 15. The LDOS for other clustersizes can be found in the Appendix. (cid:1)(cid:2) (cid:4) (cid:7) (cid:8) (cid:4) (cid:9) (cid:10) (cid:11) (cid:12)s4 (cid:8) (cid:4) (cid:9) (cid:15) (cid:16) (cid:12)s4 (cid:8) (cid:4) (cid:9) (cid:1)(cid:2)4(cid:17)(cid:1)(cid:2)4(cid:18)(cid:1)(cid:2)4(cid:19)(cid:4)4(cid:17)(cid:4)4(cid:18)(cid:20)(cid:18)(cid:19)(cid:16)(cid:21)(cid:11) 2udu2wdw21d12 (cid:1)(cid:2) u (cid:1)(cid:2) w (cid:1)(cid:2) (cid:1)(cid:2) (cid:26)o(cid:26) (cid:28)(cid:4)(cid:29)(cid:4) !- !3 !0 !w d w 0 3 FIG. 15: (Color online) The total, integrated and local den-sity of states of the Fe atoms for the Fe O +5 cluster. Thetrivalent Fe(1), Fe(2) and Fe(3) all show 3 d levels at -6 eVand small hybridization with O. The divalent Fe(4), however,shows strong hybridization and a single level at E HOMO . Table II shows the local spin moments of the clus-ters: Fe O +4 , Fe O +5 , Fe O +6 , Fe O +7 and Fe O +8 . ForFe O +4 all three Fe atoms have a similar spin momentwithin 0.04 µ B . A comparison with magnetite suggestsall Fe atoms are trivalent. This agrees with an ionic bondmodel. Furthermore, this is confirmed by the integratedand local density of states shown in Appendix A. The 3 d peaks around -6 eV correspond to 15 electrons, indicatingthe hybridization between Fe and O is small. Note that,the central oxygen atoms O(4) and O(7) are partiallyspin polarized.For Fe O +5 , the spin moment of Fe(4) is 0.5 µ B lowerthan the other Fe atoms, indicating three trivalent and asingle divalent atom. The difference is also in agreementwith the magnetite results. The Fe(4) also breaks the C symmetry as is shown in Fig. 6. The local (LDOS) andintegrated density of states are shown in Fig. 15. Notethat all Fe have 3 d peaks around − O +4 cluster. The LDOS of the divalent Fe(4) atom howevershows strong hybridization with O and a single minoritylevel at E HOMO .Whereas Fe O only contains trivalent Fe, for Fe O +6 this is no longer the case due to ionization. As can beseen from Table II, three trivalent Fe atoms are present,together with a single Fe atom. The spin moment is0reduced with respect to Fe , consistent with a higheroxidation state than Fe .In Fe O +7 , only trivalent Fe atoms are present, con-sistent with an ionic model and the ionized state of thecluster. Fe O +8 , on the other hand, is again a mixedvalence cluster where the magnetic moment of Fe(4) is0.4 µ B lower than the other Fe atoms, indicating Fe(4)is divalent. This is also consistent with the LDOS shownin Appendix A.Figure 16 shows the density of states for the differentcationic clusters and magnetite. The calculated band gapof 0.2 eV in magnetite is considerably smaller than forthe reported clusters: around 3 eV for Fe O +4 and slightlysmaller for Fe O +5 and Fe O +6 . Furthermore, whereasmagnetite has a t g orbital of Fe just below the Fermienergy, in the reported clusters Fe O +5 and Fe O +8 havea similar level due to a divalent Fe atom. Note that the 3 d orbitals of Fe in the clusters are located around 5.5 eVbelow the HOMO level, which is 2 eV higher in energycompared to magnetite. (cid:1) (cid:2)(cid:3) (cid:4) (cid:5) (cid:6) (cid:7) n (cid:9) (cid:10)n (cid:11) (cid:6) (cid:12) (cid:6) (cid:2) (cid:4) (cid:13)(cid:2) g (cid:15) -6 (cid:13)(cid:2) M (cid:15) E6 (cid:13)(cid:2)n(cid:4)(cid:13)(cid:2)n(cid:20)(cid:13)(cid:2)n(cid:21)(cid:15)n(cid:4)(cid:15)n(cid:20)n (cid:13)(cid:2) (cid:15) g6 (cid:13)(cid:2) (cid:15) M6 (cid:13)(cid:2) (cid:15) (cid:24)(cid:12)(cid:25)(cid:3)(cid:2)(cid:6)(cid:5)(cid:6)(cid:2) (cid:26)8(cid:26) (cid:28)(cid:15)(cid:24)(cid:15) n(cid:29)(cid:2)(cid:30)(cid:31) - g 3 d 7 d 3 g FIG. 16: (Color online) The density of states for Fe x O + y clus-ters. For these calculations a smearing of 0.15 eV was usedfor convenience of the reader. The HOMO level is located at0 eV and the small occupation above the HOMO level is dueto smearing. III. CONCLUSION
In this work, we have studied the geometric, elec-tronic and magnetic structure of Fe x O + y clusters usingdensity functional theory. For Fe O we compared bind-ing distances and electronic structure between the hybridB3LYP functional, and different U eff in the PBE+ U for-malism. We found the best match for U eff = 3 eV. Usingthe PBE+ U formalism and a genetic algorithm, manypossible isomers were considered. For isomers low in en-ergy, all different magnetic configurations were furthergeometrically optimized. Finally, for the cationic clus-ters we calculated the vibration spectra and comparedthem with experiments to identify the geometric struc-ture of Fe O +4 , Fe O +5 , Fe O +6 , Fe O +7 and Fe O +8 . Allcationic clusters with an even number of Fe atoms havea small magnetic moment of 1 µ B due to ionization. Fur-thermore, comparison with bulk magnetite reveals thatFe O +5 , Fe O +6 and Fe O +8 are mixed valence clusters.In contrast, in Fe O +4 and Fe O +7 all Fe are found to betrivalent. IV. ACKNOWLEDGEMENTS
The work is supported by European Research Council(ERC) Advanced Grant No. 338957 FEMTO/NANO. E. J. W. Verwey, Nature , 327 (1939). F. Walz, J. Phys.: Condens. Matter , R285 (2002). L. Andrews, G. V. Chertihin, A. Ricca, and C. W.Bauschlicher, J. Am. Chem. Soc. , 467 (1996). G. V. Chertihin, W. Saffel, J. T. Yustein, L. Andrews,M. Neurock, A. Ricca, and C. W. Bauschlicher, J. Phys.Chem. , 5261 (1996). S. Laurent, D. Forge, M. Port, A. Roch, C. Robic, L. Van-der Elst, and R. N. Muller, Chem. Rev. , 2064 (2008). N. M. Reilly, J. U. Reveles, G. E. Johnson, S. N. Khanna,and A. W. Castleman, J. Phys. Chem. A , 4158 (2007). L. S. Wang, H. Wu, and S. R. Desai, Phys. Rev. Lett. ,4853 (1996). D. Schr¨oder, P. Jackson, and H. Schwarz, Eur. J. Inorg.Chem. , 1171 (2000). A. Erlebach, H. D. Kurland, J. Grabow, F. A. M¨uller, andM. Sierka, Nanoscale , 2960 (2015). B. V. Reddy, F. Rasouli, M. R. Hajaligol, and S. N.Khanna, Fuel , 1537 (2004). B. V. Reddy and S. N. Khanna, Phys. Rev. Lett. ,068301 (2004). A. Fiedler, D. Schroeder, S. Shaik, and H. Schwarz, J. Am.Chem. Soc. , 10734 (1994). K. Ohshimo, T. Komukai, R. Moriyama, and F. Misaizu,J. Phys. Chem. A , 3899 (2014). S. Yin, W. Xue, X. L. Ding, W. G. Wang, S. G. He, andM. F. Ge, Int. J. Mass Spectrom. , 72 (2009). K. Palot´as, A. N. Andriotis, and A. Lappas, Phys. Rev. B , 075403 (2010). Q. Sun, Q. Wang, K. Parlinski, J. Z. Yu, Y. Hashi, X. G.Gong, and Y. Kawazoe, Phys. Rev. B , 5781 (2000). Q. Wang, Q. Sun, M. Sakurai, J. Z. Yu, B. L. Gu,K. Sumiyama, and Y. Kawazoe, Phys. Rev. B , 12672(1999). J. Kortus and M. R. Pederson, Phys. Rev. B , 5755(2000). Q. Sun, B. V. Reddy, M. Marquez, P. Jena, C. Gonzalez,and Q. Wang, J. Phys. Chem. C , 4159 (2007). S. L´opez, A. H. Romero, J. Mej´ıa-L´opez, J. Mazo-Zuluaga,and J. Restrepo, Phys. Rev. B , 085107 (2009). X. L. Ding, W. Xue, Y. P. Ma, Z. C. Wang, and S. G. He,J. Chem. Phys. , 014303 (2009). N. O. Jones, B. V. Reddy, F. Rasouli, and S. N. Khanna,Phys. Rev. B , 165411 (2005). H. Shiroishi, T. Oda, I. Hamada, and N. Fujima, Eur.Phys. J. D , 85 (2003). Q. Sun, M. Sakurai, Q. Wang, J. Z. Yu, G. H. Wang,K. Sumiyama, and Y. Kawazoe, Phys. Rev. B , 8500(2000). Z. Cao, M. Duran, and M. Sol`a, J. Chem. Soc., FaradayTrans. , 2877 (1998). A. Erlebach, C. H¨uhn, R. Jana, and M. Sierka, Phys.Chem. Chem. Phys. , 26421 (2014). R. L. Johnston, Dalton Trans. , 4193 (2003). M. Haertelt, A. Fielicke, G. Meijer, K. Kwapien, M. Sierka,and J. Sauer, Phys. Chem. Chem. Phys. , 2849 (2012). H. J. Zhai, J. D¨obler, J. Sauer, and L. S. Wang, J. Am.Chem. Soc. , 13270 (2007). A. Kirilyuk, A. Fielicke, K. Demyk, G. von Helden, G. Mei-jer, and T. Rasing, Phys. Rev. B , 020405 (2010). G. Kresse and J. Furthm¨uller, Phys. Rev. B , 11169(1996). P. E. Bl¨ochl, Phys. Rev. B , 17953 (1994). G. Kresse and D. Joubert, Phys. Rev. B , 1758 (1999). J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.Lett. , 3865 (1996). V. I. Anisimov and Y. Izyumov,
Electronic Structure ofStrongly Correlated Materials (Springer-Verlag Berlin Hei-delberg, 2010). V. I. Anisimov, F. Aryasetiawan, and A. I. Lichtenstein, J.Phys.: Condens. Matter , 767 (1997). A. M. Burow, T. Wende, M. Sierka, R. Wodarczyk,J. Sauer, P. Claes, L. Jiang, G. Meijer, P. Lievens, andK. R. Asmis, Phys. Chem. Chem. Phys. , 19393 (2011). S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J.Humphreys, and A. P. Sutton, Phys. Rev. B , 1505(1998). H. T. Jeng, G. Y. Guo, and D. J. Huang, Phys. Rev. Lett. , 156403 (2004). A. D. Becke, J. Chem. Phys. , 1372 (1993). G. Makov and M. C. Payne, Phys. Rev. B , 4014 (1995). J. Neugebauer and M. Scheffler, Phys. Rev. B , 16067(1992). L. Fan and T. Ziegler, J. Chem. Phys. , 9005 (1992). D. Porezag and M. R. Pederson, Phys. Rev. B , 7830(1996). J. B. Pendry, J. Phys. C , 937 (1980). M. Rossi, V. Blum, P. Kupser, G. Von Helden, F. Bierau,K. Pagel, G. Meijer, and M. Scheffler, J. Phys. Chem. Lett. , 3465 (2010). A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Phys.Rev. B , 5467 (1995). V. I. Anisimov, I. S. Elfimov, N. Hamada, and K. Terakura,Phys. Rev. B , 4387 (1996). J. P. Wright, J. P. Attfield, and P. G. Radaelli, Phys. Rev.Lett. , 266401 (2001). C. J. Cramer and D. G. Truhlar, Phys. Chem. Chem. Phys. , 10757 (2009). M. Reiher, Faraday Discuss. , 97 (2007). S. H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. ,1200 (1980). In particular, we use B3LYP with the VWN3 functional asdefined in Ref. 52.
Appendix A: Local DOS
In this appendix we show the integrated and localDOS of the clusters Fe O +4 , Fe O +6 , Fe O +7 , Fe O +8 ,and magnetite. Figures 17, 18, 19, and 20 show thetotal, integrated and local density of states of Fe O +4 ,Fe O +6 , Fe O +7 , and Fe O +8 , respectively. Of these clus-ters, Fe O +4 and Fe O +7 are pure trivalent and the LDOScontains 3 d peaks at -6 eV and small hybridization be-tween Fe and O. Fe O +6 contains a single tetravalent Featom, with a similar LDOS compared to Fe . The ion-ized electron is not removed from the 3d levels at -6 eV,but from the hybridized levels with oxygen, as can beseen from the integrated density of states. Fe O +5 andFe O +8 contain a single divalent Fe atom, which has adistinct LDOS, in which there are no peaks around -6 eV2but strong spin polarized hybridization with oxygen anda single occupied minority level at the HOMO level. Evenin bulk magnetite, as is shown in Fig. 21, the same fea-tures between divalent and trivalent Fe atoms exist. (cid:1)(cid:2) u (cid:4) w (cid:7) (cid:8) (cid:4) (cid:9) (cid:10) (cid:11) (cid:12)a3 (cid:8) (cid:4) (cid:9) (cid:15) (cid:16) (cid:12) (cid:17) (cid:18) (cid:8) (cid:4) (cid:9) (cid:1)(cid:2)3(cid:19)(cid:1)(cid:2)3(cid:20)(cid:1)(cid:2)3(cid:21)(cid:4)3(cid:19)(cid:4)3(cid:20)(cid:22)(cid:20)(cid:21)(cid:16)(cid:23)(cid:11) 5psp5dsd5 (cid:1)(cid:2) p (cid:1)(cid:2) d (cid:1)(cid:2) u (cid:28)o(cid:28) (cid:30)(cid:4)(cid:31)(cid:4) FIG. 17: (Color online) The total, integrated and local densityof states of the Fe O +4 cluster. (cid:1)(cid:2) (cid:4) (cid:7) (cid:8) (cid:4) (cid:9) (cid:10) (cid:11) (cid:12)s4 (cid:8) (cid:4) (cid:9) (cid:15) (cid:16) (cid:12)s4 (cid:8) (cid:4) (cid:9) (cid:1)(cid:2)4(cid:17)(cid:1)(cid:2)4(cid:18)(cid:1)(cid:2)4(cid:19)(cid:4)4(cid:17)(cid:4)4(cid:18)(cid:20)(cid:18)(cid:19)(cid:16)(cid:21)(cid:11) udu5wdw51d15 (cid:1)(cid:2) u (cid:1)(cid:2) w (cid:1)(cid:2) (cid:1)(cid:2) (cid:27)o(cid:27) (cid:29)(cid:4)(cid:30)(cid:4) "E "2 "0 "w d w 0 2 FIG. 18: (Color online) The total, integrated, and local den-sity of states of the Fe O +6 cluster. Fe(1) is tetravalent as isshown in Table I. (cid:1) (cid:2) (cid:3) (cid:4) (cid:5)(cid:6) (cid:3) (cid:10) (cid:11) (cid:12)sO (cid:2) (cid:3) (cid:4) (cid:15) (cid:16) (cid:12)sO (cid:2) (cid:3) (cid:4) (cid:5)(cid:6)O(cid:17)(cid:5)(cid:6)O(cid:18)(cid:5)(cid:6)O(cid:19)(cid:3)O(cid:17)(cid:3)O(cid:18)(cid:20)(cid:18)(cid:19)(cid:16)(cid:21)(cid:11) udwd1d0d2d (cid:5)(cid:6) u (cid:5)(cid:6) w (cid:5)(cid:6) (cid:5)(cid:6) (cid:5)(cid:6) (cid:27)o(cid:27) (cid:29)(cid:3)(cid:30)(cid:3) O(cid:31)(cid:6) ! "E "3 "0 "w d w 0 3
FIG. 19: (Color online) The total, integrated, and local den-sity of states of the Fe O +7 cluster. TABLE II: The spin moment for Fe x O + y clusters. The atom numbers correspond to the atom numbers shown in Figures 4,6,10,11, and 13. The spin moment is calculated using atomic spheres of 1.3 and 0.82 ˚A for Fe and O, respectively.Cluster Spin moment [ µ B ]1 2 3 4 5 6 7 8Fe O +4 Fe − .
84 3 .
88 3 .
88O 0 .
56 0 .
00 0 .
00 0 . O +5 Fe 3 . − .
84 3 . − . − .
05 0 .
13 0 . − .
05 0 . O +6 Fe − .
22 3 .
85 3 . − .
79O 0 .
01 0 .
54 0 . − .
25 0 .
00 0 . O +7 Fe 3 .
85 3 .
87 3 . − . − .
80O 0 .
01 0 .
10 0 .
03 0 . − .
09 0 .
05 0 . O +8 Fe 3 . − .
84 3 . − . − .
84 3 .
88O 0 .
01 0 .
51 0 .
01 0 . − .
10 0 . − .
10 0 . (cid:1)(cid:2) (cid:4) (cid:7) (cid:8) (cid:4) (cid:9) (cid:10) (cid:11) (cid:12)s6 (cid:8) (cid:4) (cid:9) (cid:15) (cid:16) (cid:12)s6 (cid:8) (cid:4) (cid:9) (cid:1)(cid:2)6(cid:17)(cid:1)(cid:2)6(cid:18)(cid:1)(cid:2)6(cid:19)(cid:4)6(cid:17)(cid:4)6(cid:18)(cid:20)(cid:18)(cid:19)(cid:16)(cid:21)(cid:11) udwd1d0d2d (cid:1)(cid:2) u (cid:1)(cid:2) w (cid:1)(cid:2) (cid:1)(cid:2) (cid:1)(cid:2) (cid:1)(cid:2) (cid:28)o(cid:28) (cid:30)(cid:4)(cid:31)(cid:4) FIG. 20: (Color online) The total, integrated, and local den-sity of states of the Fe O +8 cluster. All Fe atoms are trivalentexcept for Fe(4), which is divalent. (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:6)(cid:5) (cid:8) (cid:9) (cid:10) (cid:11) (cid:12) (cid:13) (cid:6)3g (cid:9) (cid:10) (cid:11) (cid:16)(cid:5) S(cid:20)T(cid:16)(cid:5) ). S(cid:23)AT(cid:16)(cid:5) S(cid:23))T(cid:16)(cid:5) S(cid:23)2T(cid:16)(cid:5) ). S(cid:23)BT (cid:26)F(cid:26) (cid:16) g(cid:28)(cid:5)(cid:29)(cid:30) (cid:31)- (cid:31)4 (cid:31)B (cid:31)) ( ) B 4
FIG. 21: (Color online) The total and local density of statesof the different Fe atoms in magnetite. The numbering isconsistent with Table I. Fe and Fe3+